View of Effect of optimum stratification on sampling with varying probabilities under proportional allocation

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EFFECT OF OPTIMUM STRATIFICATION ON SAMPLING WITH VARYING PROBABILITIES UNDER PROPORTIONAL ALLOCATION S.E.H. Rizvi, J.P. Gupta, M. Bhargava

1.INTRODUCTION

In stratified random sampling, proper choice of strata boundaries is one of the important factors as regards to the efficiency of estimator of population charac-teristic under consideration. The problem of optimum stratification for univariate case was first considered by Dalenius (1950) who treated the estimation variable itself as stratification variable. Singh and Sukhatme (1969) provided some ap-proximate solutions for the strata boundaries by using an auxiliary variable as the stratification variable under Neyman and proportional allocations. Singh and Sukhatme (1972) also considered this problem in case the units from different strata are selected with probability proportional to the value of auxiliary variable and with replacement (PPSWR) using Neyman allocation. In the sequel, Singh (1975) tackled the same problem for proportional and equal allocation proce-dures. In the former case they observed a gain in efficiency, however little, whereas in the latter case their study showed a remarkable decrease in the effi-ciency as compared to unstratified PPSWR.

Generally, information on more than one study variable is collected in any sur-vey and in that case the theory concerning the construction of strata for one study variable can not be adopted as such. For such a situation, Rizvi et al. (2000) developed the theory of optimum stratification for bivariate case, on the basis of auxiliary variable, in case of simple random sampling. Thus, in the present paper the problem of optimum stratification has been considered for two study vari-ables in case the units within a stratum are selected using PPSWR scheme. In this connection, the minimal equations have been developed under proportional allo-cation along with its approximate solutions. A limit expression of the generalized variance is also provided which enables us to examine the effect of stratification on sampling with PPSWR as regards to the change in number of strata. The pa-per concludes with numerical illustrations for three density functions.

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2.VARIANCE AND COVARIANCE UNDER SUPER-POPULATION SET-UP

Let the population of size N be divided into L strata and the units within a stratum are selected using PPSWR. Let the variables under study be denoted by

Yj ( j = 1, 2) and the known auxiliary variable by X. Let Phi denote the selection probability assigned to the i-th unit in the h-th stratum, i = 1, 2,..., Nh; h = 1, 2, ...,

L so that _hi

i

P 1

¦

for each h, Phi = xhi/Xh where xhi is the i-th sampled observa-tion in the h-th stratum and Xh is the stratum total for X.

The unbiased estimators of the population means Y of the study variables Yj j ( j = 1, 2) will be the weighted means defined by

hi jhi n 1 = i h L 1 = h Pj st P y n N 1 = y h

¦

1 . ( j =1, 2)

where symbols have their usual meanings.

For the proportional method of allocation, the terms for the variances of the unbiased estimators yst Pj. ( j = 1, 2) and their covariance, under repeated random

sampling are given by

2 . 1 V h L N jhi ₂ jh st Pj P 2 h=1 h i=1 hi y 1 ) ( y = _{- Y} n_N W P ª º « » « » ¬ ¼

¦

(1) and 1 2 1 2 . 1 . 2 . 1 Cov h L N hi hi h h st P st P P 2 h=1 h i=1 hi y y 1 ) ( y , y = _{- Y Y} n_N W P ª º « » ¬ ¼

¦

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where Wh is the proportion of units in the h-th stratum.

Let us assume that the population under consideration is finite and is a random sample of size N from an infinite superpopulation with same characteristics as those of the finite population. Let the joint density function of (X,Y1,Y2) in the super-population set-up be denoted by fs(x,y1,y2) and the marginal density function of X by f(x) assuming that these functions are continuous in the range (a,b) of X. Further, let us assume that the regression of Yj on X, in each stratum of the

su-per-population, is linear and that these regression lines pass through origin. Thus the regression models are given by

Yj = ƢjhX + ej ( j = 1, 2) (3)

where ej is the error component such that E(ej~X) = 0, E(ejej'~x,x') = 0 for xzx' and V(ej~X) = Ƨj(x)>0 ( j =1,2) for all x(a, b).

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2 h N jhi 2 2 jh jh i=1 hi y = _{- Y} P V

¦

, ( j =1, 2) (4) and 12 1 2 1 2 . h N hi hi h h h i=1 hi y y = _{- Y Y} P V

¦

. (5) The expectations of 2 1h V , 2 2h

V and V12h in the superpopulation become 1 2

E(V2jh) =E E (V2jh), ( j = 1, 2) and E(V12h) =E E1 2(V12h) respectively.

Here, E1 is the expectation over all possible x's and E2 is the expectation for fixed

(x1, x2, ..., xN). Thus, from (4) we have

2 2 2 2 E E E h N jhi 2 2 jh jh i=1 hi y ( ) = - (_Y ) P V ª« º» « » ¬

¦

¼ . (6)

And, using model (3) we get

2 2 2 2 E E E h h 2 N N hi jhi jh 2 jh jh hi jhi i=1 hi i=1 ( x +e ) ( ) = - ( x +e ) P E E V ª« º» ª_« º_» « » ¬ ¼ ¬

¦

¼

¦

which on simplification becomes

' 2 ' E h N 2 hi hi jh j i i (V ) = I (_x )_x z

¦

(7) where I j(xhi) =K j(xhi)/xhi .

Now taking overall expectation, we get

1 2 E(V2jh) =E E (V2jh) ' 1 ' E h N hi hi j i i = I (x )x z

¦

.

Since i-th and i'-th units in the h-th stratum are drawn independently, we have E j 2 jh h h h hx (V ) =_{N N}( - 1)P P_I (8) where j hI

P and P_hx are the expected values of Ij(x) and x, respectively, in the

h-th stratum. Now, assuming the stratum size Nh to be large enough so that

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E

j

2 2

jh h h hx

(V _{) = N} P P_I . (9)

And, from (5) we have

1 2 2 2 2 1 2 1 . E E E h N hi hi h h 12h i hi y y ( ) = - (_{Y Y} ) P V ª_« º_» ¬

¦

¼ . (10)

Using the model (3) the above expression can be put as

2 2 { } E E h N h 12h 1h hi 1hi 2h hi 2hi hi i=1 (V ) =X «ª (E x +e )(E x +e )/x º» ¬

¦

¼ E2 h h N N hi 1hi hi 2hi 1h 2h i=1 i=1 (E x +e ) ( E x +e ) ª º ª º « » « » ¬

¦

¼ ¬

¦

¼

which, on some algebraic simplifications, gives E2(Ƴ12h) = 0.

Therefore, E (Ƴ12h) = 0. (11)

Thus, the expected variances and covariance of the estimators yst Pj. ( j = 1, 2),

under proportional method of allocation, become

2 . V j L h j st Pj P h hx h=1 1 ) ( y = _W n I V

¦

P P (12) and Cov( y_{st P}_{. 1}, y_{st P}_{. 2} )_P= 0. (13)

The minimal equations and its solutions have been obtained in the next sec-tion.

3.MINIMAL EQUATIONS

For the division of the population under consideration into L strata, let us rep-resent the set of the points of demarcation forming L strata by {xh}, h = 1, 2, ...,

L-1. These points are obtained by solving the minimal equations which are

ob-tained by equating to zero the partial derivatives of G3w.r.t. {xh}, where

G3 = 12 1 21 2 2 2 V V V _V . (14) Hence, we get

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2 1 12 1 2 12 2 2 2 2 h h h + - 2 = 0 x x x V V V V w V w V w w w w . (15)

Putting the values of 2 j

V and V12 from (12) and (13), respectively, in (15), we

have 1 2 2 2 1 1 L L h h hx h h hx r r rx h h hx h h hx r r rx h h=1 h h=1 W ( + )+ ( + )=0 W W W W W x x I I I I I I P P w P P P P P P P P P P w § · _§ _{· w} ¨ ¸ ¨ ¸ ¨ ¸ _© _¹_w ©

¦

¹

¦

(16) where r = h+1, h = 1,2,...,L-1.

The values of partial derivatives of the various terms involved in (16) can be easily obtained on the lines of Singh and Sukhatme (1969). Putting these values in (16), we finally get the minimal equations as follows:

{ } { } 1 2 2 2 1 1 L L h h h h h h hx h hx 2 h h h hx h hx 1 h h=1 h=1 W P PI x PI +P I( (x )-PI ) + W P PI x PI+P I( (x )-PI) § · § · ¨ ¸ ¨ ¸ ©

¦

¹ ©

¦

¹ { } { } 1 2 2 2 1 1 L L h h h h h h hx r rx 2 r h h hx r rx 1 r h=1 h=1 W W =¨§ P PI ¸· x PI +P I( (x )-PI ) +¨§ P PI ·¸ x PI+P I( (x )-PI) ©

¦

¹ ©

¦

¹ (17) where r = h+1, h = 1, 2,..., L-1.

The solution of this system of minimal equations will provide the set of the optimum points of stratification {xh} corresponding to the minimum value of the

generalized variance G3. But, the system of equations given by (17) involves

pa-rameters which are themselves functions of points of strata boundaries indicating thereby that the exact solutions are difficult to obtain. Hence, some approximate solutions have been obtained which are given in the next section.

4.APPROXIMATE SOLUTIONS OF THE MINIMAL EQUATIONS

In order to obtain the approximate solutions of the minimal equations (17), we have to expand both the sides of the minimal equation about the point xh, the

common boundary point of the h-th and r-th strata. For this purpose, we impose certain regularity conditions on the functions f(x),Ij(x) and Ƨj(x) as under:

A function ƹ(x) belongs to the class of function, ƙ, if it satisfies the following conditions:

(i) 0 <ƹ(x) < f,

and (ii) ƹ(x), ƹ'(x) and ƹ''(x) exist and are continuous for all x (a,b), where (b-a) < f.

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Under these assumptions, we can obtain the series expansions for Wh, j

hI

P

and P_hx by using Taylor's theorem about both the upper and lower boundaries of the h-th stratum. Proceeding on the lines of Singh and Sukhatme (1969), the expanded forms of different terms involved in the left hand side of the minimal equations (17) can be obtained as follows:

2 3 4 h h h h h h f f f = f 1- K + - + O( ) W K K K K 2f 6f 24f c cc ccc ª º « » ¬ ¼ (18) ' ' '' ' '' ''' ' j 2 2 j j j ₂ j j j j ₃ ₄ h h h h j h 2 j j j f f f + 2f ff + ff + -= 1- K + K - K +O(K ) 2 12f 24f I I I I I I I I P I I I I ª _c _cc _c _c º « » « » ¬ ¼ (19) where Kh is the width of h-th stratum, the function Ij, f and their derivatives are

evaluated at x = xh .

From (19), taking Ij(t) = t, we have

2 2 3 4 h h h h hx 2 f f ff -1 = x 1- K + K - K + O(K ) 2x 12fx 24f x P ª_« c cc c º_» ¬ ¼. (20)

Now, multiplying (18), (19) and (20) and then summing over all the strata we shall get, by proceeding on the lines of Singh and Sukhatme (1969), the following use-ful expression: h j j h-1 x L L 2 2 h h hx h j h h=1 h=1 _x 1 = - ( t ) f ( t )dt [1 O(K )] W K 12 I K P P P P

¦

_{¦ ³}

(21) where j b j a = (t)f(t)dt K P

³

K and I_j(t) =K _j(t)/t .

For finding out approximate solutions to the minimal equations (17), as ob-tained in the preceding section, we shall first expand the expressions on the left hand side of the minimal equation, and then the right hand side may be expanded similarly.

On using series expansions of

j

hI

P and Phx from (19) and (20), respectively, and the result (21), the left hand side of the minimal equations (17) can be put as

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1 2 2 2 1 2 ' ' '' ' ' h h-1 x L 2 2 h 2 h 2 h h h h=1 _x f + f 1 K K C 1- K +O(K ) - (t)f(t)dt[ 1+O(K )] 12 4 3 _f K I I I I P I ª º ª _c º « »« » « » « » ¬ ¼¬

¦ ³

¼ 1 1 1 2 1 ' ' '' ' ' h 2 h-1 x L 2 2 h 2 h 2 h h h h=1 _x f + f 1 K K + 1- K +O(K ) - (t)f(t)dt[ 1+O(K ) 12 4 3 _f K

]

I I I I P I ª º ª _c º « » « » « » « » ¬ ¼¬

¦ ³

¼ (22) where 2 1 L L h h 1 h hx 2 h hx h=1 h=1 C = xI

¦

_W P P_I + xI

¦

_W P P_I .

On simplifying (22) and neglecting the terms of O(m4_{), m =}_{Sup K}

h, we can put

the left hand side of (17), after some algebraic simplifications, as

2 2' 1' h 1 h-1 2/3 1/3 _x 2 2 3 h h x [f( + ) ] C - K l (t)f(t)dt[1+ O(K ) 4f

]

KI KI P P ª º « » « » ¬

³

¼ (23) where l3(t) =P_K₁I2'(t) +P_K₂I1'(t). (24)

Similarly the right hand side of the minimal equation can be shown to be equal to 2 2' 1' h 1 h-1 2/3 1/3 _x 2 2 r 3 r x [f( + ) ] C - K l (t)f(t)dt[1+ O(K ) 4f

]

KI KI P P ª º « » « » ¬

³

¼ . (25)

Substituting (23) and (25) in (17) and then equating both the sides, we get the following equalities 2 2' 1' h 1 h-1 2/3 1/3 _x 2 2 3 h h x [f( + ) ] (t)f(t)dt[1+ O( ) l K K 4f

]

KI KI P P ª º « » « » ¬

³

¼ 2 2' 1' h 1 h-1 2/3 1/3 _x 2 2 r 3 r x [f( + ) ] (t)f(t)dt[1+ O( ) l K K 4f

]

KI KI P P ª º « » « » ¬

³

¼ (26)

where l3(t) is as given by (24), r = h+1, h =1,2,...,L-1 and we assume that the

func-tion l3(t)f(t) ƙ, the class of all functions f(x), such that x (a ,b) where (b-a) < f

and the first two derivatives of f(x) exist for all x in the interval (a ,b).

Now, if we assume that the number of strata is large enough so that the strata widths Kh are small and higher powers of Kh in the expansion can be neglected,

then the system of minimal equations (17) and equivalently the equations (26) can approximately be put as

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h h+1 h-1 h 2/3 2/3 x x 2 2 3 h 3 r x x (t)f(t)dt = (t)f(t)dt l l K K ª º ª º « » « » « » « » ¬

³

¼ ¬

³

¼ (27) which gives ₃ h h-1 x 2 h x l (t)f(t)dt K

³

= constant, h=1, 2,...,L (28) or equivalently by Q3(xh-1,xh) = constant (29)

where Q3(xh-1,xh) is a function of order O(m3), Sup( )

(a,b) h m K such that 3 h h-1 x 2 2 h 3 h-1 h h x l (t)f(t)dt = Q (_x ,_x )[1+ O( )] K

³

K . (30)

Various methods of finding approximate solutions to the minimal equations (17) can be established through the system of equations (28). Singh and Sukhatme (1969) developed different forms of Q(xh-1,xh) corresponding to univariate case

under Neyman allocation. Proceeding on the same lines, one such form of the function Q3(xh-1,xh) can be obtained as

h h-1 3 x 3 3 x (t)f(t)dt l ª º « » « » ¬

³

¼ = constant (say, C) (31) where 3 3 b 3 3 a 1 C = l (t)f(t)dt L ª º « » « » ¬

³

¼ .

We now propose the following rule of finding out approximately optimum strata boundaries (AOSB), in case of two study variables, when the method of varying probabilities of selection is adopted.

Cum 3 _R_3(x)_rule:

If the function R3(x) = l3(x)f(x), is bounded and possesses its first two

deriva-tives, then for a given value of L taking equal intervals on the cumulative cube root of R3(x), that is h h-1 b x 3 3 3 3 a x 1 (x)dx = (x)dx R R L

³

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5.EFFECT OF OPTIMUM STRATIFICATION

In this section, we shall discuss as to whether stratification provides any gain on using PPSWR method of selecting samples from different strata, under pro-portional method of allocation. For this purpose a limiting expression of general-ized variance G3 as defined by (14) is to be obtained. Now, under the regression

model (3), the generalized variance G3 can be expressed as

1 2 L L 2 h 3 h h hx h hx h=1 h=1 = W n G P PI W P PI § ·§ · ¨ ¸¨ ¸ ©

¦

¹©

¦

¹. (32)

Further, on using result 21, the above expression can be put in its expanded form as 2 1 3 h j h-1 x L 2 2 h h h=1 _x 1 n G = - K (t)f(t)dt[1+ O(K ) 12

]

K P Ic ª º « » « » ¬

¦ ³

¼ 2 1 2 2 2 1 1 1 12 h h-x L h h Ƨ h= _x µ - K ƶ (t)f(t)dt[ + O(K )

]

ª º c u« » « » ¬

¦ ³

¼ (33)

which on neglecting the terms of order O(m4_), _{Sup (} ₎

( , )

h m K

a b

, can be obtained as 2 h 1 h-1 x L 2 2 2 3 3 h h h=1 _x 1 = - (t)f(t)dt[1+ O( ) n G K l K 12

]

K K P P

¦ ³

. (34)

For further simplifying (34), we have the following lemma due to Singh and Suk-hatme (1969), which can be proved by expanding O f(t) about the point t = xh on

making use of Taylor's theorem.

Lemma 5.1. h h h-1 h-1 x x -1 2 h h x x f(t) dt =K f(t) dt[1+ O(K )] O O O ª º « » « » ¬

³

¼

³

. (35)

Now, by using lemma 5.1 in (34), we get

2 h 1 h-1 3 x L 2 3 3 3 h=1 x 1 = - (t)f(t)dt n G l 12 K K P P ª« º» « » ¬ ¼

¦ ³

. (36)

In case the strata boundaries are determined by making use of the proposed 3

3(x)

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2 3= - ₂ n G 12L E D or, 3 ₂ ₂ 1 = -G 12 n L E D ª º « » ¬ ¼ (37) where 1 2 = _K _K D P P and . 3 b 3 3 a = l (t)f(t)dt E ª_« º_» « » ¬

³

¼

Taking limit on both the sides of (37) as Lof, we get

3 2 G n

lim

L D of . (38)

From relation (38) we infer that the generalized variance G3 will be less than

ơ/n2_{, which goes on increasing with the increase in number of strata and} ap-proaches its maximum value ơ/n2_{. This result leads to the following remarks:}

1. The limiting expression (38) is exactly same as the result (38) of Rizvi et al. (2000), indicating thereby that the stratified simple random sampling is equally efficient as PPSWR when the number of strata becomes large, under proportional method of allocation.

2. Here, it may be pointed out that the result (38) is also supported by Suk-hatme et al. (1984) wherein it is observed that the efficiency of stratified PPSWR would decrease as the allocation departs from Neyman allocation. 3. It is interesting to note that the similar observation have also been made by

Singh (1975), for univariate case, under proportional allocation.

The empirical investigations in this regard have been made in the next section.

6.EMPIRICAL STUDY

The effectiveness of the proposed method of finding the set {xh} of AOSB has

been demonstrated empirically. For this purpose the following three density func-tions of the stratification variable X have been considered.

1. Uniform distribution: f(x) = 1 1dxd2

2. Right triangular distribution: f(x) = 2(2-x) 1dxd2 3. Exponential distribution: f(x) = e-x+1 _1dxdf

These densities to some extent represent those usually encountered in practice. For the purpose of comparison of the proposed method with stratified simple random sampling (SRS), we have taken the same specifications and procedures as given by Rizvi et al. (2000). The point of truncation as well as the form of condi-tional variance i.e. ( ) gj

j x A xj

K , ( j = 1, 2), where Aj > 0 and gj = 0, 1, 2, are

also the same.

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1, 2 and g1 = g2 = 2 could be taken as the other combinations have no signifi-cance. For instance, for g1= g2= 0 or g1= g2= 1, the function l3(t), as defined by

(4.7), becomes zero, and for such cases any set of boundaries is optimum.

For the aforesaid combinations, values of AOSB, n2_G₃_{and per cent relative}

ef-ficiency of proposed method with unstratified PPSWR (% RE1) and per cent rela-tive efficiency of stratified PPSWR with stratified SRS (% RE2) have been pre-sented in tables 1, 2 and 3 for uniform, right triangular and exponential distribu-tions, respectively. These values indicate that the percent relative efficiency goes on decreasing with the increase in number of strata, however slowly, as compared to unstratified PPSWR sampling scheme for all the three distributions considered. This trend is obvious from the relation (36) which is also supported by the nu-merical investigations made by Singh (1975) for univariate case. Also in each case the relative efficiency goes on decreasing with the increase in the value of g1 or g2 and attains minimum value for g1= g2= 2. Similar finding has also been reported

by Singh (1975). Through perusal of the following tables it may be inferred that the % RE1, in case of right triangular distribution, is slightly higher than that for the uniform distribution. In the former case it varies from 97.79 to 94.17 and that for the latter case it ranges between 97.27 to 93.13. When the stratification vari-able follows exponential distribution, the percent increase in the relative effi-ciency was found to be less than the other two distributions.

TABLE 1

AOSB and relative efficiencies of optimum stratification (Uniform distribution)

AOSB L g1, g2 = 1, 2 n2G3 %RE1 %RE2 1 1.00000 2.00000 .0066944 100.00 623.13 2 1.00000 1.49653 2.00000 .0068822 97.27 226.19 3 1.00000 1.32990 1.65981 2.00000 .0069177 96.77 155.61 4 1.00000 1.24649 1.49298 1.73947 .0069287 96.62 131.74 2.00000 5 1.00000 1.19648 1.39296 1.58945 .0069361 96.51 120.33 1.78593 2.00000 6 1.00000 1.16318 1.32635 1.48953 .0069388 96.48 114.21 1.65271 1.81588 2.00000 g1, g2 = 2 1 1.00000 2.00000 .0064554 100.00 646.21 2 1.00000 1.49723 2.00000 .0068200 94.65 228.26 3 1.00000 1.33050 1.66101 2.00000 .0068898 93.69 158.09 4 1.00000 1.24719 1.49438 1.74157 .0069128 93.38 132.04 2.00000 5 1.00000 1.19718 1.39436 1.59155 .0069258 93.21 120.56 1.78873 2.00000 6 1.00000 1.16388 1.32775 1.49163 .0069314 93.13 114.33 1.65551 1.81939 2.00000

If we look at %RE2 values, it is found to be as high as 862.59% for exponential distribution followed by right triangular distribution (638.10%) and uniform distri-bution (646.21%). A further comparison of proposed method with that of Rizvi et

al. (2000) reveals that %RE2 goes on decreasing as the number of strata increases. And, it is remarkable to note that for larger number of strata the stratified SRS would found to be equally efficient as PPSWR when proportional method of allo-cation is envisaged. This fact is also supported by remark 2 under section 5.

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TABLE 2

AOSB and relative efficiencies of optimum stratification (Right triangular distribution)

AOSB L g1 , g2 = 1, 2 n2G3 %RE1 %RE2 1 1.00000 2.00000 .0029920 100.00 618.75 2 1.00000 1.40142 2.00000 .0030597 97.79 242.49 3 1.00000 1.25849 1.55249 2.00000 .0030741 97.33 165.55 4 1.00000 1.19048 1.39747 1.63235 .0030797 97.15 137.55 2.00000 5 1.00000 1.15057 1.31065 1.48443 .0030825 97.06 124.29 1.68111 2.00000 6 1.00000 1.12427 1.25484 1.39351 .0030842 97.01 117.00 1.54379 1.71337 2.00000 g1 , g2 = 2 1 1.00000 2.00000 .0029013 100.00 638.10 2 1.00000 1.40222 2.00000 .0030337 95.63 244.57 3 1.00000 1.25929 1.55439 2.00000 .0030621 94.75 166.20 4 1.00000 1.19118 1.39907 1.63525 .0030727 94.42 137.87 2.00000 5 1.00000 1.15127 1.31225 1.48703 .0030780 94.26 124.47 1.68521 2.00000 6 1.00000 1.12507 1.25644 1.39602 .0030810 94.17 117.12 1.54749 1.71877 2.00000 TABLE 3

AOSB and relative efficiencies of optimum stratification (Exponential distribution)

AOSB L g1 , g2 = 1, 2 n2G3 %RE1 %RE2 1 1.00000 6.00000 .5944715 100.00 710.35 2 1.00000 2.55953 6.00000 .6812319 87.26 270.39 3 1.00000 1.94507 3.33092 6.00000 .7027103 84.60 178.33 4 1.00000 1.67921 2.55797 3.80547 .7108992 83.62 144.73 6.00000 5 1.00000 1.53028 2.17468 2.99579 .7148522 83.16 128.86 4.12998 6.00000 6 1.00000 1.43499 1.94388 2.55654 .7170472 82.90 120.13 3.32717 4.36679 6.00000 g1 , g2 = 2 1 1.00000 6.00000 .4895521 100.00 862.59 2 1.00000 2.55991 6.00000 .6428539 76.15 286.53 3 1.00000 1.94532 3.33180 6.00000 .6840050 71.57 183.21 4 1.00000 1.67946 2.55872 3.80710 .7000186 69.93 146.98 6.00000 5 1.00000 1.53053 2.17531 2.99692 .7077993 69.16 130.14 4.13223 6.00000 6 1.00000 1.43511 1.94426 2.55729 .7121312 68.74 120.96 3.32855 4.36941 6.00000

Division of Agricultural Economics & Statistics SYED EJAZ HUSAIN RIZVI S.K. University of Agricultural Sciences and Technology

Main Campus, Chatha, Jamnu, India

Meerut City (U.P.) India JAI P. GUPTA Department of Agronomy MANOJ BHARGAVA HPKV, Palampur, HP, India

ACKNOWLEDGEMENT

The authors wish to thank the referee for his constructive suggestions and valuable comments.

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REFERENCES

T. DALENIUS, (1950), The problem of optimum stratification, “Skandinavisk Aktuaritidskrift”, 33,

pp. 203-213.

S.E.H. RIZVI, J.P. GUPTA, R. SINGH (2000), Approximately optimum stratification for two study variables using auxiliary information, “Journal of the Indian Society of Agricultural Statistics”, 53, pp. 287-298.

R. SINGH (1975), A note on optimum stratification in sampling with varying probabilities, “Australian

Journal of Statistics”, 17, pp. 12-21.

R. SINGH, B.V. SUKHATME (1969), Optimum stratification, “Annals of the Institute of Statistical

Mathematics”, 21, pp. 515-528.

R. SINGH, B.V. SUKHATME (1972), Optimum stratification in sampling with varying probabilities,

“An-nals of the Institute of Statistical Mathematics”, 24, pp. 485-494.

P.V. SUKHATME, B.V. SUKHATME, S. SUKHATME, C. ASHOK (1984), Sampling theory of surveys with ap-plications, 3rd edition, Indian Society of Agricultural Statistics, New Delhi, India, pp. 175-176.

RIASSUNTO

Effetto della stratificazione nel campionamento a probabilità variabile nel caso di allocazione proporzio-nale

Il problema della stratificazione ottima di una variabile ausiliare quando le unità dei dif-ferenti strati sono scelte con probabilità proporzionale al valore della variabile ausiliare è stata considerata da Singh (1975) nel contesto univariato. In questo lavoro viene esteso il metodo al caso di due variabili, proponendo una regola per ottenere un’approssimazione dei limiti degli strati. Attraverso sviluppi teorici e applicativi viene mostrato che l’uso della stratificazione ha effetti inversi sulla efficienza relativa rispetto alla non-stratificazione.

SUMMARY

Effect of optimum stratification on sampling with varying probabilities under proportional allocation

The problem of optimum stratification on an auxiliary variable when the units from different strata are selected with probability proportional to the value of auxiliary variable (PPSWR) was considered by Singh (1975) for univariate case. In this paper we have ex-tended the same problem, for proportional allocation, when two variates are under study.

A cum. 3

3(x)

R rule for obtaining approximately optimum strata boundaries has been

provided. It has been shown theoretically as well as empirically that the use of stratifica-tion has inverse effect on the relative efficiency of PPSWR as compared to unstratified PPSWR method when proportional method of allocation is envisaged. Further compari-son showed that with increase in number of strata the stratified simple random sampling is equally efficient as PPSWR.